#6814912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,25)(16,21)(17,23)(18,22)(19,24)(20,27)(26,41)(28,39)(29,36)(30,37)(31,38)(32,40)(33,35)(34,43)(42,44);(5,7)(6,9)(8,12)(10,11)(13,15,18,27)(14,20,22,25)(16,29,30,33)(17,32,28,19)(21,35,37,36)(23,24,39,40)(26,42,43,31)(34,44,41,38);(1,2,3,4)(13,16,19,31,18,30,32,42)(14,21,24,38,22,37,40,44)(15,26,28,29,27,43,17,33)(20,34,23,35,25,41,39,36);(6,10)(9,11)(14,22)(16,30)(20,25)(23,39)(24,40)(26,43)(29,33)(31,42);(7,12)(9,11)(13,17,32,27,18,28,19,15)(14,23,40,20,22,39,24,25)(16,29,42,26,30,33,31,43)(21,36,44,41,37,35,38,34)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722108);